98 LIMITS AND FLUXIONS 



limita the circumference. The process here tran- 

 scends our power of imagination, but lies withiii the 

 limits of reason. We are dwelling upon this point 

 because of its extreme importance in the history of 

 the theory of Hmits. 



II 8. Robins constructs upon his first definition 

 the theorem, " that, when varying magnitudes keep 

 constantly the same proportion to each other, their 

 ultimate magnitudes are in the same proportion." 

 As a coroUary of this he states **that the ultimate 

 magnitudes of the same or equal varying magni- 

 tudes are equal." 



II. Definition: " If there be tvvo quantities, 

 that are (one or both) continually varying, either 

 by being continually augmented, or continually 

 diminished ; though the proportion, they bear to 

 each other, should by this means perpetually vary, 

 but in sudi a manner, that it constantly approaches 

 nearer and nearer to some determined proportion, 

 and can also be brought at length in its approach 

 nearer to this determined proportion than to any 

 other, that can be assigned, but can never pass it : 

 this determined proportion is then called the ulti- 

 mate proportion, or the ultimate ratio of those 

 varying quantities." 



Theorem: **To this definition of the sense, in 

 which the term ultimate ratio, or ultimate pro- 

 portion is to be understood, we must subjoin the 

 following proposition : That ali the ultimate ratios 

 of the same varying ratio are the same with each 

 other." 



